Fractional view of heat‐like equations via the Elzaki transform in the settings of the Mittag–Leffler function

Rashid, Saima; Kubra, Khadija Tul; Abualnaja, Khadijah M.

doi:10.1002/mma.7793

Cited by 10 publications

(7 citation statements)

References 56 publications

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“…We have the following 5th-order approximate solution in the original space for Equations (40) and (41) when ω = 1 uses the procedure mentioned in Subsection 2.1:…”

Section: Numerical Examplesmentioning

confidence: 99%

“…These transformations are paired with additional analytical, numerical, or homotopy-based techniques to handle FODEs [30][31][32][33][34][35]. Numerous mathematicians have recently become interested in a transformation known as the Elzaki transform (ET) [36][37][38][39][40][41]. The ET was introduced by Elzaki to facilitate the process of solving ordinary and partial DEs in the time domain [42].…”

Section: Introductionmentioning

confidence: 99%

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A Novel Numerical Technique for Fractional Ordinary Differential Equations with Proportional Delay

Liaqat

Khan

Akgül

et al. 2022

Journal of Function Spaces

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Some researchers have combined two powerful techniques to establish a new method for solving fractional-order differential equations. In this study, we used a new combined technique, known as the Elzaki residual power series method (ERPSM), to offer approximate and exact solutions for fractional multipantograph systems (FMPS) and pantograph differential equations (PDEs). In Caputo logic, the fractional-order derivative operator is measured. The Elzaki transform method and the residual power series method (RPSM) are combined in this novel technique. The suggested technique is based on a new version of Taylor’s series that generates a convergent series as a solution. Establishing the coefficients for a series, like the RPSM, necessitates computing the fractional derivatives each time. As ERPSM just requires the concept of a zero limit, we simply need a few computations to get the coefficients. The novel technique solves nonlinear problems without the need for He’s and Adomian polynomials, which is an advantage over the other combined methods based on homotopy perturbation and Adomian decomposition methods. The relative, recurrence, and absolute errors of the problems are analyzed to evaluate the efficiency and consistency of the presented method. Graphical significances are also identified for various values of fractional-order derivatives. As a result, the procedure is quick, precise, and easy to implement, and it yields outstanding results.

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“…We have the following 5th-order approximate solution in the original space for Equations (40) and (41) when ω = 1 uses the procedure mentioned in Subsection 2.1:…”

Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Novel Numerical Technique for Fractional Ordinary Differential Equations with Proportional Delay

Liaqat

Khan

Akgül

et al. 2022

Journal of Function Spaces

View full text Add to dashboard Cite

show abstract

“…The ADM is a semi-analytical approach to solving linear-nonlinear FDEs by advantageously creating a functional series solution, initially presented by Adomian [48]. Later, this approach was used with numerous transformations (such as the Sumudu, Aboodh, Laplace, and Mohand transforms), as shown in [49][50][51][52][53][54][55][56][57][58].…”

Section: Introductionmentioning

confidence: 99%

Analytic Fuzzy Formulation of a Time-Fractional Fornberg–Whitham Model with Power and Mittag–Leffler Kernels

et al. 2021

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This manuscript assesses a semi-analytical method in connection with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). Moreover, we use the aforesaid strategy to address the time-fractional Fornberg–Whitham equation (FWE) under gH-differentiability by employing different initial conditions (IC). Several algebraic aspects of the fuzzy Caputo fractional derivative (CFD) and fuzzy Atangana–Baleanu (AB) fractional derivative operator in the Caputo sense, with respect to the Elzaki transform, are presented to validate their utilities. Apart from that, a general algorithm for fuzzy Caputo and AB fractional derivatives in the Caputo sense is proposed. Some illustrative cases are demonstrated to understand the algorithmic approach of FWE. Taking into consideration the uncertainty parameter ζ∈[0,1] and various fractional orders, the convergence and error analysis are reported by graphical representations of FWE that have close harmony with the closed form solutions. It is worth mentioning that the projected approach to fuzziness is to verify the supremacy and reliability of configuring numerical solutions to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures.

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“…The versatility of HPM allows it to yield approximate and exact solutions to both linear and nonlinear problems without the necessity for discretization and linearization, as with analytical methods [45]. Various studies have extensively used the HPM to analyze linear and nonlinear PDEs [46][47][48].…”

Section: Introductionmentioning

confidence: 99%

Novel Numerical Investigations of Fuzzy Cauchy Reaction–Diffusion Models via Generalized Fuzzy Fractional Derivative Operators

et al. 2021

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The present research correlates with a fuzzy hybrid approach merged with a homotopy perturbation transform method known as the fuzzy Shehu homotopy perturbation transform method (SHPTM). With the aid of Caputo and Atangana–Baleanu under generalized Hukuhara differentiability, we illustrate the reliability of this scheme by obtaining fuzzy fractional Cauchy reaction–diffusion equations (CRDEs) with fuzzy initial conditions (ICs). Fractional CRDEs play a vital role in diffusion and instabilities may develop spatial phenomena such as pattern formation. By considering the fuzzy set theory, the proposed method enables the solution of the fuzzy linear CRDEs to be evaluated as a series of expressions in which the components can be efficiently identified and generating a pair of approximate solutions with the uncertainty parameter λ∈[0,1]. To demonstrate the usefulness and capabilities of the suggested methodology, several numerical examples are examined to validate convergence outcomes for the supplied problem. The simulation results reveal that the fuzzy SHPTM is a viable strategy for precisely and accurately analyzing the behavior of a proposed model.

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Fractional view of heat‐like equations via the Elzaki transform in the settings of the Mittag–Leffler function

Cited by 10 publications

References 56 publications

A Novel Numerical Technique for Fractional Ordinary Differential Equations with Proportional Delay

A Novel Numerical Technique for Fractional Ordinary Differential Equations with Proportional Delay

Analytic Fuzzy Formulation of a Time-Fractional Fornberg–Whitham Model with Power and Mittag–Leffler Kernels

Novel Numerical Investigations of Fuzzy Cauchy Reaction–Diffusion Models via Generalized Fuzzy Fractional Derivative Operators

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